I am not a specialist of number theory, so please excuse my ignorance: is the following question still an open problem?
Let $k \in \mathbb{N}^*$, are there infinitely many prime numbers of the form $n^{2^k}+1$?

Your question is still open. It is a special case of Schinzel's Hypothesis H applied to the polynomial $f(x)=x^{2^k}+1$.

As Bjorn mentions in his comment, the case of $k=1$ is a particularly famous unsolved problem. It is the fourth of Landau's problems (Edmund Landau was a famous German number theorist during the early twentieth century).

Anyway by Friedlander and Iwaniec (1997). They proved that there are infinitely many primes of the form $x^2 + y^4 .$ They mention near the end that they do not have a proof for primes of the form $x^2 + y^6 $ but would like one. So there is a way to go to settle $x^2 + 1.$

FYI, what I did (not remembering title, authors, anything but the result) was write a program to give the primes $x^2 + y^4 $ and put the first dozen in Sloane's sequence site search feature.